It is not difficult to build new manifolds out of old in the smooth category, for example

 - taking the direct product or constructing a fiber bundle,
 
 - taking the level set of a regular value of a smooth function,
 
 - forming the *connected sum* and more generally gluing (in different ways) two manifolds

The connected sum construction is very flexible and important for the topological classification of manifolds.
Unfortunately, in the complex category it doesn't behave well *: maybe the connected sum doesn't admit a complex structure at all.
In my attempt to understand the topology of complex manifolds, it is natural to ask 

>What are the most common/powerful constructions that allows us to form a new complex manifold $X = \mathcal{F}(M, N)$ given two complex manifolds $M,N$?
 
The constructions I am interested in should have a topological interpretation and when meaningful the complex structure on $X$ should be compatible with the one of $M,N$.




*see https://mathoverflow.net/questions/216272/does-bbbcp2n-bbbcp2n-ever-support-an-almost-complex-structure and  https://math.stackexchange.com/questions/1411694/is-the-connected-sum-of-complex-manifolds-also-complex)